b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 15.1s

Alternatives: 12

Speedup: 1.0×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- 1.0 m) (+ (/ (* m (- 1.0 m)) v) -1.0)))

C

double code(double m, double v) {
	return (1.0 - m) * (((m * (1.0 - m)) / v) + -1.0);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (1.0d0 - m) * (((m * (1.0d0 - m)) / v) + (-1.0d0))
end function

Java

public static double code(double m, double v) {
	return (1.0 - m) * (((m * (1.0 - m)) / v) + -1.0);
}

Python

def code(m, v):
	return (1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)

Julia

function code(m, v)
	return Float64(Float64(1.0 - m) * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
end

MATLAB

function tmp = code(m, v)
	tmp = (1.0 - m) * (((m * (1.0 - m)) / v) + -1.0);
end

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \]
4. Add Preprocessing

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -1}{v} - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (+ -1.0 (/ m v)) (+ -1.0 (- 2.0 m)))
   (* m (- (* m (/ (+ m -1.0) v)) -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * ((m * ((m + -1.0) / v)) - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = ((-1.0d0) + (m / v)) * ((-1.0d0) + (2.0d0 - m))
    else
        tmp = m * ((m * ((m + (-1.0d0)) / v)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * ((m * ((m + -1.0) / v)) - -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m))
	else:
		tmp = m * ((m * ((m + -1.0) / v)) - -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(-1.0 + Float64(m / v)) * Float64(-1.0 + Float64(2.0 - m)));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(Float64(m + -1.0) / v)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	else
		tmp = m * ((m * ((m + -1.0) / v)) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(2.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(N[(m + -1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-undefine 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \]

      3. log1p-undefine 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(e^{\color{blue}{\log \left(1 + \left(1 - m\right)\right)}} + \left(-1\right)\right) \]

      4. rem-exp-log 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\color{blue}{\left(1 + \left(1 - m\right)\right)} + \left(-1\right)\right) \]

      5. associate-+r- 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\color{blue}{\left(\left(1 + 1\right) - m\right)} + \left(-1\right)\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\left(\color{blue}{2} - m\right) + \left(-1\right)\right) \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\left(2 - m\right) + \color{blue}{-1}\right) \]

   6. Simplified 99.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(\left(2 - m\right) + -1\right)} \]

   7. Taylor expanded in m around 0 97.2%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(\left(2 - m\right) + -1\right) \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

      3. associate-/l* 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in m around inf 98.1%

      \[\leadsto \color{blue}{\left(-1 \cdot m\right)} \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 98.1%

         \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -1}{v} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (+ -1.0 (/ m v)) (+ -1.0 (- 2.0 m)))
   (* m (+ 1.0 (/ (* m (+ m -1.0)) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = ((-1.0d0) + (m / v)) * ((-1.0d0) + (2.0d0 - m))
    else
        tmp = m * (1.0d0 + ((m * (m + (-1.0d0))) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m))
	else:
		tmp = m * (1.0 + ((m * (m + -1.0)) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(-1.0 + Float64(m / v)) * Float64(-1.0 + Float64(2.0 - m)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * Float64(m + -1.0)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (-1.0 + (m / v)) * (-1.0 + (2.0 - m));
	else
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(2.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-undefine 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \]

      3. log1p-undefine 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(e^{\color{blue}{\log \left(1 + \left(1 - m\right)\right)}} + \left(-1\right)\right) \]

      4. rem-exp-log 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\color{blue}{\left(1 + \left(1 - m\right)\right)} + \left(-1\right)\right) \]

      5. associate-+r- 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\color{blue}{\left(\left(1 + 1\right) - m\right)} + \left(-1\right)\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\left(\color{blue}{2} - m\right) + \left(-1\right)\right) \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(\left(2 - m\right) + \color{blue}{-1}\right) \]

   6. Simplified 99.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(\left(2 - m\right) + -1\right)} \]

   7. Taylor expanded in m around 0 97.2%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(\left(2 - m\right) + -1\right) \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

      3. associate-/l* 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{\left(-1\right)}\right) \]

      2. sub-neg 99.9%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v} - 1\right)} \]

      3. clear-num 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\frac{1}{\frac{v}{1 - m}}} - 1\right) \]

      4. un-div-inv 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} - 1\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m}{\frac{v}{1 - m}} - 1\right)} \]

   7. Taylor expanded in m around inf 98.0%

      \[\leadsto \color{blue}{\left(-1 \cdot m\right)} \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \]
   8. Step-by-step derivation

      1. neg-mul-1 98.1%

         \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \]

   9. Simplified 98.0%

      \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \]

   10. Taylor expanded in m around 0 98.0%

       \[\leadsto \color{blue}{m \cdot \left(1 + m \cdot \left(\frac{m}{v} - \frac{1}{v}\right)\right)} \]
   11. Step-by-step derivation

       1. sub-neg 98.0%

          \[\leadsto m \cdot \left(1 + m \cdot \color{blue}{\left(\frac{m}{v} + \left(-\frac{1}{v}\right)\right)}\right) \]

       2. distribute-lft-in 55.9%

          \[\leadsto m \cdot \left(1 + \color{blue}{\left(m \cdot \frac{m}{v} + m \cdot \left(-\frac{1}{v}\right)\right)}\right) \]

       3. distribute-neg-frac 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + m \cdot \color{blue}{\frac{-1}{v}}\right)\right) \]

       4. metadata-eval 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + m \cdot \frac{\color{blue}{-1}}{v}\right)\right) \]

       5. associate-/l* 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \color{blue}{\frac{m \cdot -1}{v}}\right)\right) \]

       6. *-commutative 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \frac{\color{blue}{-1 \cdot m}}{v}\right)\right) \]

       7. associate-*r/ 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \color{blue}{-1 \cdot \frac{m}{v}}\right)\right) \]

       8. distribute-rgt-in 98.1%

          \[\leadsto m \cdot \left(1 + \color{blue}{\frac{m}{v} \cdot \left(m + -1\right)}\right) \]

       9. associate-*l/ 98.0%

          \[\leadsto m \cdot \left(1 + \color{blue}{\frac{m \cdot \left(m + -1\right)}{v}}\right) \]

   12. Simplified 98.0%

       \[\leadsto \color{blue}{m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (- 1.0 m) (+ -1.0 (/ m v)))
   (* m (+ 1.0 (/ (* m (+ m -1.0)) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * (1.0d0 + ((m * (m + (-1.0d0))) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * (1.0 + ((m * (m + -1.0)) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * Float64(m + -1.0)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * (1.0 + ((m * (m + -1.0)) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0 97.2%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

      3. associate-/l* 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{\left(-1\right)}\right) \]

      2. sub-neg 99.9%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v} - 1\right)} \]

      3. clear-num 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\frac{1}{\frac{v}{1 - m}}} - 1\right) \]

      4. un-div-inv 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} - 1\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m}{\frac{v}{1 - m}} - 1\right)} \]

   7. Taylor expanded in m around inf 98.0%

      \[\leadsto \color{blue}{\left(-1 \cdot m\right)} \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \]
   8. Step-by-step derivation

      1. neg-mul-1 98.1%

         \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \]

   9. Simplified 98.0%

      \[\leadsto \color{blue}{\left(-m\right)} \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \]

   10. Taylor expanded in m around 0 98.0%

       \[\leadsto \color{blue}{m \cdot \left(1 + m \cdot \left(\frac{m}{v} - \frac{1}{v}\right)\right)} \]
   11. Step-by-step derivation

       1. sub-neg 98.0%

          \[\leadsto m \cdot \left(1 + m \cdot \color{blue}{\left(\frac{m}{v} + \left(-\frac{1}{v}\right)\right)}\right) \]

       2. distribute-lft-in 55.9%

          \[\leadsto m \cdot \left(1 + \color{blue}{\left(m \cdot \frac{m}{v} + m \cdot \left(-\frac{1}{v}\right)\right)}\right) \]

       3. distribute-neg-frac 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + m \cdot \color{blue}{\frac{-1}{v}}\right)\right) \]

       4. metadata-eval 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + m \cdot \frac{\color{blue}{-1}}{v}\right)\right) \]

       5. associate-/l* 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \color{blue}{\frac{m \cdot -1}{v}}\right)\right) \]

       6. *-commutative 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \frac{\color{blue}{-1 \cdot m}}{v}\right)\right) \]

       7. associate-*r/ 55.9%

          \[\leadsto m \cdot \left(1 + \left(m \cdot \frac{m}{v} + \color{blue}{-1 \cdot \frac{m}{v}}\right)\right) \]

       8. distribute-rgt-in 98.1%

          \[\leadsto m \cdot \left(1 + \color{blue}{\frac{m}{v} \cdot \left(m + -1\right)}\right) \]

       9. associate-*l/ 98.0%

          \[\leadsto m \cdot \left(1 + \color{blue}{\frac{m \cdot \left(m + -1\right)}{v}}\right) \]

   12. Simplified 98.0%

       \[\leadsto \color{blue}{m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot \left(m + -1\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - m\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} + -1\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- 1.0 m) (+ (/ m (/ v (- 1.0 m))) -1.0)))

C

double code(double m, double v) {
	return (1.0 - m) * ((m / (v / (1.0 - m))) + -1.0);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (1.0d0 - m) * ((m / (v / (1.0d0 - m))) + (-1.0d0))
end function

Java

public static double code(double m, double v) {
	return (1.0 - m) * ((m / (v / (1.0 - m))) + -1.0);
}

Python

def code(m, v):
	return (1.0 - m) * ((m / (v / (1.0 - m))) + -1.0)

Julia

function code(m, v)
	return Float64(Float64(1.0 - m) * Float64(Float64(m / Float64(v / Float64(1.0 - m))) + -1.0))
end

MATLAB

function tmp = code(m, v)
	tmp = (1.0 - m) * ((m / (v / (1.0 - m))) + -1.0);
end

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

   2. sub-neg 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{\left(-1\right)}\right) \]

   2. sub-neg 99.8%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v} - 1\right)} \]

   3. clear-num 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\frac{1}{\frac{v}{1 - m}}} - 1\right) \]

   4. un-div-inv 99.9%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} - 1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m}{\frac{v}{1 - m}} - 1\right)} \]

7. Final simplification 99.9%

   \[\leadsto \left(1 - m\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} + -1\right) \]
8. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- 1.0 m) (+ (* m (/ (- 1.0 m) v)) -1.0)))

C

double code(double m, double v) {
	return (1.0 - m) * ((m * ((1.0 - m) / v)) + -1.0);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (1.0d0 - m) * ((m * ((1.0d0 - m) / v)) + (-1.0d0))
end function

Java

public static double code(double m, double v) {
	return (1.0 - m) * ((m * ((1.0 - m) / v)) + -1.0);
}

Python

def code(m, v):
	return (1.0 - m) * ((m * ((1.0 - m) / v)) + -1.0)

Julia

function code(m, v)
	return Float64(Float64(1.0 - m) * Float64(Float64(m * Float64(Float64(1.0 - m) / v)) + -1.0))
end

MATLAB

function tmp = code(m, v)
	tmp = (1.0 - m) * ((m * ((1.0 - m) / v)) + -1.0);
end

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

   2. sub-neg 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ -1.0 (/ m v)) (* (/ m v) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m / v) * (m + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m / v) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(m + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = -1.0 + (m / v);
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

      3. associate-/l* 99.7%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0 96.8%

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

   6. Taylor expanded in v around 0 97.1%

      \[\leadsto \color{blue}{\frac{m}{v}} - 1 \]

   if 2.39999999999999991 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Taylor expanded in m around inf 0.1%

      \[\leadsto \color{blue}{\frac{m}{v}} \cdot \left(1 - m\right) \]
   5. Step-by-step derivation

      1. sub-neg 0.1%

         \[\leadsto \frac{m}{v} \cdot \color{blue}{\left(1 + \left(-m\right)\right)} \]

      2. pow1 0.1%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \color{blue}{{\left(-m\right)}^{1}}\right) \]

      3. metadata-eval 0.1%

         \[\leadsto \frac{m}{v} \cdot \left(1 + {\left(-m\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \]

      4. sqrt-pow1 76.7%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \color{blue}{\sqrt{{\left(-m\right)}^{2}}}\right) \]

      5. pow2 76.7%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \sqrt{\color{blue}{\left(-m\right) \cdot \left(-m\right)}}\right) \]

      6. sqr-neg 76.7%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \sqrt{\color{blue}{m \cdot m}}\right) \]

      7. sqrt-prod 76.7%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \color{blue}{\sqrt{m} \cdot \sqrt{m}}\right) \]

      8. add-sqr-sqrt 76.7%

         \[\leadsto \frac{m}{v} \cdot \left(1 + \color{blue}{m}\right) \]

      9. distribute-lft-in 76.7%

         \[\leadsto \color{blue}{\frac{m}{v} \cdot 1 + \frac{m}{v} \cdot m} \]

      10. *-commutative 76.7%

          \[\leadsto \color{blue}{1 \cdot \frac{m}{v}} + \frac{m}{v} \cdot m \]

      11. *-un-lft-identity 76.7%

          \[\leadsto \color{blue}{\frac{m}{v}} + \frac{m}{v} \cdot m \]

   6. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{m}{v} + \frac{m}{v} \cdot m} \]
   7. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \frac{m}{v} + \color{blue}{m \cdot \frac{m}{v}} \]

      2. distribute-rgt1-in 76.7%

         \[\leadsto \color{blue}{\left(m + 1\right) \cdot \frac{m}{v}} \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\left(m + 1\right) \cdot \frac{m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-1 + \frac{m}{v}\right) \cdot \left(m + 1\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (+ -1.0 (/ m v)) (+ m 1.0)))

C

double code(double m, double v) {
	return (-1.0 + (m / v)) * (m + 1.0);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = ((-1.0d0) + (m / v)) * (m + 1.0d0)
end function

Java

public static double code(double m, double v) {
	return (-1.0 + (m / v)) * (m + 1.0);
}

Python

def code(m, v):
	return (-1.0 + (m / v)) * (m + 1.0)

Julia

function code(m, v)
	return Float64(Float64(-1.0 + Float64(m / v)) * Float64(m + 1.0))
end

MATLAB

function tmp = code(m, v)
	tmp = (-1.0 + (m / v)) * (m + 1.0);
end

Wolfram

code[m_, v_] := N[(N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Add Preprocessing

3. Taylor expanded in m around 0 46.7%

   \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
4. Step-by-step derivation

   1. sub-neg 46.7%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)} \]

   2. flip-+ 46.7%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\frac{1 \cdot 1 - \left(-m\right) \cdot \left(-m\right)}{1 - \left(-m\right)}} \]

   3. metadata-eval 46.7%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \frac{\color{blue}{1} - \left(-m\right) \cdot \left(-m\right)}{1 - \left(-m\right)} \]

   4. clear-num 46.7%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\frac{1}{\frac{1 - \left(-m\right)}{1 - \left(-m\right) \cdot \left(-m\right)}}} \]

   5. un-div-inv 46.7%

      \[\leadsto \color{blue}{\frac{\frac{m}{v} - 1}{\frac{1 - \left(-m\right)}{1 - \left(-m\right) \cdot \left(-m\right)}}} \]

   6. sub-neg 46.7%

      \[\leadsto \frac{\color{blue}{\frac{m}{v} + \left(-1\right)}}{\frac{1 - \left(-m\right)}{1 - \left(-m\right) \cdot \left(-m\right)}} \]

   7. metadata-eval 46.7%

      \[\leadsto \frac{\frac{m}{v} + \color{blue}{-1}}{\frac{1 - \left(-m\right)}{1 - \left(-m\right) \cdot \left(-m\right)}} \]

   8. clear-num 46.7%

      \[\leadsto \frac{\frac{m}{v} + -1}{\color{blue}{\frac{1}{\frac{1 - \left(-m\right) \cdot \left(-m\right)}{1 - \left(-m\right)}}}} \]

   9. metadata-eval 46.7%

      \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{\frac{\color{blue}{1 \cdot 1} - \left(-m\right) \cdot \left(-m\right)}{1 - \left(-m\right)}}} \]

   10. flip-+ 46.7%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{\color{blue}{1 + \left(-m\right)}}} \]

   11. sub-neg 46.7%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{\color{blue}{1 - m}}} \]

   12. sub-neg 46.7%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{\color{blue}{1 + \left(-m\right)}}} \]

   13. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{1 + \color{blue}{\sqrt{-m} \cdot \sqrt{-m}}}} \]

   14. sqrt-prod 86.4%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{1 + \color{blue}{\sqrt{\left(-m\right) \cdot \left(-m\right)}}}} \]

   15. sqr-neg 86.4%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{1 + \sqrt{\color{blue}{m \cdot m}}}} \]

   16. sqrt-prod 86.4%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{1 + \color{blue}{\sqrt{m} \cdot \sqrt{m}}}} \]

   17. add-sqr-sqrt 86.4%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{1 + \color{blue}{m}}} \]

   18. +-commutative 86.4%

       \[\leadsto \frac{\frac{m}{v} + -1}{\frac{1}{\color{blue}{m + 1}}} \]

5. Applied egg-rr 86.4%

   \[\leadsto \color{blue}{\frac{\frac{m}{v} + -1}{\frac{1}{m + 1}}} \]
6. Step-by-step derivation

   1. associate-/r/ 86.4%

      \[\leadsto \color{blue}{\frac{\frac{m}{v} + -1}{1} \cdot \left(m + 1\right)} \]

   2. /-rgt-identity 86.4%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right)} \cdot \left(m + 1\right) \]

   3. *-commutative 86.4%

      \[\leadsto \color{blue}{\left(m + 1\right) \cdot \left(\frac{m}{v} + -1\right)} \]

   4. +-commutative 86.4%

      \[\leadsto \left(m + 1\right) \cdot \color{blue}{\left(-1 + \frac{m}{v}\right)} \]

7. Simplified 86.4%

   \[\leadsto \color{blue}{\left(m + 1\right) \cdot \left(-1 + \frac{m}{v}\right)} \]

8. Final simplification 86.4%

   \[\leadsto \left(-1 + \frac{m}{v}\right) \cdot \left(m + 1\right) \]
9. Add Preprocessing

Alternative 9: 62.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-154}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 8.5e-154) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-154) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.5d-154) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-154) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.5e-154:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.5e-154)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.5e-154)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.5e-154], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.4999999999999996e-154

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

      3. associate-/l* 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in m around 0 79.4%

      \[\leadsto \color{blue}{-1} \]

   if 8.4999999999999996e-154 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0 29.7%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]

   4. Taylor expanded in m around inf 22.7%

      \[\leadsto \color{blue}{\frac{m}{v}} \cdot \left(1 - m\right) \]

   5. Taylor expanded in m around 0 54.4%

      \[\leadsto \color{blue}{\frac{m}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ -1 + \frac{m}{v} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (+ -1.0 (/ m v)))

C

double code(double m, double v) {
	return -1.0 + (m / v);
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (-1.0d0) + (m / v)
end function

Java

public static double code(double m, double v) {
	return -1.0 + (m / v);
}

Python

def code(m, v):
	return -1.0 + (m / v)

Julia

function code(m, v)
	return Float64(-1.0 + Float64(m / v))
end

MATLAB

function tmp = code(m, v)
	tmp = -1.0 + (m / v);
end

Wolfram

code[m_, v_] := N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

   2. sub-neg 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in m around 0 70.7%

   \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]

6. Taylor expanded in v around 0 70.8%

   \[\leadsto \color{blue}{\frac{m}{v}} - 1 \]

7. Final simplification 70.8%

   \[\leadsto -1 + \frac{m}{v} \]
8. Add Preprocessing

Alternative 11: 27.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (+ m -1.0))

C

double code(double m, double v) {
	return m + -1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = m + (-1.0d0)
end function

Java

public static double code(double m, double v) {
	return m + -1.0;
}

Python

def code(m, v):
	return m + -1.0

Julia

function code(m, v)
	return Float64(m + -1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = m + -1.0;
end

Wolfram

code[m_, v_] := N[(m + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

   2. sub-neg 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 26.4%

   \[\leadsto \color{blue}{-1 \cdot \left(1 - m\right)} \]
6. Step-by-step derivation

   1. neg-mul-1 26.4%

      \[\leadsto \color{blue}{-\left(1 - m\right)} \]

   2. neg-sub0 26.4%

      \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

   3. associate--r- 26.4%

      \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

   4. metadata-eval 26.4%

      \[\leadsto \color{blue}{-1} + m \]

7. Simplified 26.4%

   \[\leadsto \color{blue}{-1 + m} \]

8. Final simplification 26.4%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 12: 24.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 -1.0)

C

double code(double m, double v) {
	return -1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = -1.0d0
end function

Java

public static double code(double m, double v) {
	return -1.0;
}

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

function tmp = code(m, v)
	tmp = -1.0;
end

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]

   2. sub-neg 99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in m around 0 23.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))